The parabolic algebra revisited

The parabolic algebra A p is the weakly closed operator algebra on L 2 ( ℝ ) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions e i λ x , λ ≥ 0 . It is reflexive, with an invariant subspace lattice Lat A p which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of Lat A p is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson’s notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that Lat A p is not synthetic relative to the H ∞ ( ℝ ) subalgebra of A p . Also, various new operator algebras, derived from isometric representations and from compact perturbations of A p , are defined and identified.